NATIONAL BOARD FOR HIGHER MATHEMATICS. M. A. and M.Sc. Scholarship Test. September 19, Time Allowed: 150 Minutes Maximum Marks: 30
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1 NATIONAL BOARD FOR HIGHER MATHEMATICS M. A. and M.Sc. Scholarship Test September 19, 2015 Time Allowed: 150 Minutes Maximum Marks: 30 Please read, carefully, the instructions on the following page 1
2 INSTRUCTIONS TO CANDIDATES Please ensure that this question paper booklet contains 8 numbered (and printed) pages. The reverse of each printed page is blank and can be used for rough work. There are three parts to this test: Algebra, Analysis and Miscellaneous. Each part consists of 10 questions adding up to 30 questions in all. Answer each question, as directed, in the space provided for it in the answer booklet, which is being supplied separately. This question paper is meant to be retained by you and so do not answer questions on it. In certain questions you are required to pick out the qualifying statement(s) from multiple choices. None of the statements, or one or more than one statement may qualify. Write none if none of the statements qualify, or list the labels of all the qualifying statements (amongst (a),(b) and (c)). Points will be awarded in the above questions only if all the correct choices are made. There will be no partial credit. Notations N denotes the set of natural numbers {1, 2, 3, }, Z - the integers, Q - the rationals, R - the reals and C - the field of complex numbers. R n denotes the n-dimensional Euclidean space. The symbol Z n will denote the ring of integers modulo n. ( ) n The symbol will denote the standard binomial coefficient r giving the number of ways of choosing r objects from a collection of n objects, where n 1 and 0 r n are integers. The symbol ]a, b[ will stand for the open interval {x R a < x < b} while [a, b] will stand for the corresponding closed interval; [a, b[ and ]a, b] will stand for the corresponding left-closed-rightopen and left-open-right-closed intervals respectively. We denote by M n (R) (respectively, M n (C)), the set of all n n matrices with entries from R (respectively, C). The trace of a square matrix A will be denoted tr(a) and the determinant by det(a). The derivative of a function f will be denoted by f and the second derivative by f. All logarithms, unless specified otherwise, are to the base e. Calculators are not allowed. 2
3 Section 1: Algebra 1.1 Which of the following statements are true? a. If G is a group such that (ab) 2 = a 2 b 2 for all a, b G, then G is abelian. b. If G is a group such that a 2 = e for all a G, where e is the identity element in G, then G is abelian. c. If G is a group such that a 2 = e for all a G, where e is the identity element in G, then G is finite. 1.2 For real numbers a and b, define the mapping τ a,b : R R by τ a,b (x) = ax + b. Let G = {τ a,b : a, b R, a 0}. Under composition of mappings, this becomes a group. Which of the following subgroups of G are normal? a. H = {τ a,b a 0, a Q, b R}. b. H = {τ 1,b b R}. c. H = {τ 1,b b Q}. 1.3 Find all positive integers n > 1 such that the polynomial x 4 + 3x 3 + x 2 + 6x + 10 belongs to the ideal generated by the polynomial x 2 + x + 1 in Z n [x]. 1.4 Find the number of irreducible monic polynomials of degree 2 over the field F 5 of five elements. 1.5 Find the number of invertible 2 2 matrices with entries in F 2, the field of two elements. 1.6 Let V be a finite dimensional vector space and let A, B and C be subspaces of V. Which of the following statements are true? a. A (B + C) = A B + A C. b. A (B + C) A B + A C. c. A (B + C) A B + A C. 1.7 Write down the inverse of the following matrix: Find an orthogonal matrix P such that P AP 1 = B, where A = and B =
4 1.9 Let A = (a ij ) M n (R), where a ij = { 1 if i + j = n + 1, 0 otherwise. What is the value of det(a) when (i) n = 10 and (ii) n = 100? 1.10 Let n > 1 be a positive integer and let A M n (R) be as defined in Problem 1.9 above. Write down the set of eigenvalues of A. 4
5 Section 2: Analysis 2.1 In each of the following cases, state whether the given series is absolutely convergent, conditionally convergent or divergent. a b. c. 2.2 Evaluate: ( 1) n n n + 1. n=1 n=2 ( 1) n 1 n(n 1). lim (1 2 tan θ 0 θ)cot θ. 2.3 Let f : R R be continuous. Let t R. Evaluate: 2.4 Evaluate: 2.5 Define: f(x) = 1 lim h 0 h lim n t+h t h n k=1 f(s) ds. n n 2 + k 2. { an + sin πx if x [2n, 2n + 1], b n + cos πx if x ]2n 1, 2n[, where n varies over N {0}. Find all possible sequences {a n } and {b n } such that f is continuous on [0, [. 2.6 Which of the following functions are uniformly continuous on the prescribed domain? a. f(x) = x sin 1 on ]0, 1[. x b. f(x) = e x cos 1 on ]0, 1[. x c. f(x) = sin(sin x) on ]0, [. 2.7 Let k be a positive integer. Let a > 0. Evaluate: (a + 1 n lim )n (a + 2 n )n (a + k n )n. n a nk 2.8 Which of the following functions defined on R are differentiable? a. f(x) = x x. b. f(x) = [x] sin 2 πx. c. f(x) = x. (The symbol [x] in statement (b) stands for the integral part of x R, i.e. the largest integer less than, or equal to, x.) 5
6 2.9 Which of the following statements are true? a. Let f be continuously differentiable in [a, b] and twice differentiable on ]a, b[. If f(a) = f(b) and if f (a) = 0, then there exists x 0 ]a, b[ such that f (x 0 ) = 0. b. Let f be continuously differentiable in [a, b]. If f(a) = f(b) and if f (a) = f (b), then there exists x 1 and x 2 in ]a, b[ such that x 1 x 2 and such that f (x 1 ) = f (x 2 ). c. Let f be continuously differentiable on [0, 2], and twice differentiable on ]0, 2[. If f(0) = 0, f(1) = 1 and f(2) = 2, then there exists x 0 ]0, 2[ such that f (x 0 ) = Let Γ be the boundary of the square in the complex plane with vertices at the points 0, 1, 1+i and i, which is described in the anticlockwise direction. Evaluate: πe πz dz. Γ 6
7 Section 3: Miscellaneous 3.1 Which of the following relations are true? a. ( 1) n(n 1) 2 = ( 1) n(n+1) 2. b. ( 1) n(n 1) 2 = ( 1) [ n 2 ]. c. ( 1) n(n 1) 2 = ( 1) n2. (The symbol [ n] in statement (b) stands for the integral part of n ; see also 2 2 Problem 2.8.) 3.2 Let n, k and r be positive integers such that k < r < n and also such that n > r + k. Which of the following statements are true? a. ( ) n = n ( ) n 1. r r r 1 b. c. ( ) n r ( ) n r = = k l=0 l=0 ( k l ) ( n k r l ). k ( ) ( ) k n l. l r k 3.3 Let a, b, c R. Evaluate the determinant: a b c 2a 2a 2b b c a 2b 2c 2c c a b. 3.4 Find the sum of the following series up to n terms: Let N = Find the number of divisors of N, excluding 1 and N. 3.6 Let P n be the polygon in the complex plane with vertices situated at the n-th roots of unity. Let A n be its area and let L n be its perimeter. Evaluate: L 2 n lim. n A n 3.7 Let a, b R. For P = (x, y) R 2, define f(p ) = ax + by. The line ax + by = 4 intersects the line segment joining two points P 1 and P 2 in the plane at a point R such that P 1 R : RP 2 = 1 : 2. If f(p 1 ) = 3, what is the value of f(p 2 )? 3.8 Let n N. Which of the following statements are true? a. x n 1 n(x 1)x n 1 2 for all x 0. b. x n 1 n(x 1)x n 1 2 for all x 1. c. x n 1 n(x 1)x n 1 2 for all x n. 7
8 3.9 What is the radius of the sphere with centre at the origin and which has the plane x + y + z = 1 as a tangent? 3.10 Let S = {(x, y) R 2 x 2 + 2hxy + y 2 = 1}. For what values of h is the set S non-empty and bounded? 8
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